# Decision with Multiple Criteria

• Michel Grabisch
Chapter
Part of the Theory and Decision Library C book series (TDLC, volume 46)

## Abstract

This second application chapter explores decision with multiple criteria, usually called “multicriteria decision making” (MCDM). It deals with situations where the decision maker has to make decision considering together several points of view (criteria), which are often antagonistic. This covers many everyday life decision problems, like choosing a restaurant or a movie, buying a new car or renting an apartment, etc. Our presentation of the topic is unconventional, although based on classical concepts and results from multiattribute utility theory (MAUT) and measurement theory. We start from scratch, and ask ourselves under which conditions does the decomposable model (a very commonly used model consisting in assigning numerical scores on each criterion and aggregating them into a single overall score), exist, and how to build it. While the answer to the first question (conditions of existence) is well known and is a standard result of measurement theory, the answer to the second question is less obvious, and it is precisely here that we are unconventional. We show that in order to build scores on criteria, either difference measurement can be applied (under the assumption of weak difference independence), or reference points must be found on each criterion, which permits to apply the MACBETH method. With the latter method, we show that we are naturally lead to the use of the Choquet integral for aggregating the scores of the criteria. Interestingly enough, with the former approach based on the assumption of weak difference independence, it is known since the seventies through the works of Dyer and Sarin, Keeney and Raiffa, that the only possible model of aggregation is the multilinear model, also called Owen extension of a capacity. Remembering that the Choquet integral is the Lovász extension of a capacity, this shows in a striking way that capacities are firmly rooted in multicriteria decision making.

## Keywords

Binary Relation Reference Level Multiple Criterion Aggregation Function Numerical Representation
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